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Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture. (English) Zbl 1277.11072

Summary: Let \(E/ \mathbb Q\) be an elliptic curve and \(K/ \mathbb Q\) a finite Galois extension with group \(G\). We write \(E_K\) for the base change of \(E\) and consider the equivariant Tamagawa number conjecture for the pair \((h^1(E_K)(1),\mathbb Z[G])\). This conjecture is an equivariant refinement of the Birch and Swinnerton-Dyer conjecture for \(E/K\). For almost all primes \(l\), we derive an explicit formulation of the conjecture that makes it amenable to numerical verifications. We use this to provide convincing numerical evidence in favor of the conjecture.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G05 Elliptic curves over global fields

Citations:

Zbl 1277.11071
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